Asymptotics for a wave equation with critical exponential nonlinearity

In this paper, we discuss some qualitative analysis of solutions to the following Cauchy problem of wave equations involving the 1/2-Laplace operator with critical exponential nonlinearity {u(tt) + (-Delta)(1/2) u + delta u(t) + u = lambda vertical bar u vertical bar(q-2) ue(alpha 0u2) in R x (0, +infinity), u(x, 0) = u(0)(x), u(f)(x, 0) = u(1)(x) in R, where lambda > 0, delta >= 0, q > 2, and alpha(0) > 0. By using the contraction mapping principle, we show that the above Cauchy problem has a unique local solution. With the help of the potential well argument, we characterize the stable sets by the asymptotic behavior of solutions as t goes to infinity, as well as the unstable sets by the blow-up of solutions in finite time.